A Course on Relativity and Electrodynamics

A few notes on a course on “Relativity and Electrodynamics” that I have taught twice so far. This is a core course for third-year undergraduates and is also taken by master’s and IPhD students. It was the first full course that I taught after joining here. The students are assumed to have already taken a first course on Electromagnetism, as well as Classical Mechanics (Lagrangian and Hamiltonian formulations).

First of all, I was very pleased to see that the prescribed textbook was The Classical Theory of Fields. The course is also designed around the text. Of course, the aim was not to cover each and every section of the book, especially given that it is Landau–Lifshitz. My aim, through the lectures and the detailed notes I provided, was to unpack the rich insights hidden within the terse style of the text. For me, this course comes at a juncture where students have completed the basic courses of the first two years and have begun — or at least should begin — to take physics more seriously. This course contains certain broad and central themes of physics to which students can readily be exposed.

Here are some of the things that I tried to highlight, while doing my best to keep the language and articulation simple:

1. Dispelling misconceptions about relativity
   Much of the popular, and even standard pedagogical, discourse on relativity proceeds along the lines of “everything is relative” and “observer-dependent,” with emphasis on length contraction, time dilation, and similar effects. It was stressed in the class that the heart of relativity lies in the invariance of the laws of physics under Lorentz transformations, and in the central role played by frame-independent quantities. This shifts the focus toward invariant structures and quantities shared across all inertial frames. The notions of covariance and invariance were stressed and demonstrated repeatedly. The rest follows from them.

2. Special relativity as spacetime geometry
   Special relativity is already a theory about the geometry of spacetime, albeit a flat one. It already reveals the causal structure of spacetime. Sometimes this point is lost amidst all the discussion about frames and observers.

3. The significance of finite propagation speed
   There is much discussion about light being special because of its speed. Conceptually, however, what is more important is the existence of a universal finite speed for the propagation of interactions. This is a recurring theme throughout the course and serves as the logical precursor to many later concepts, such as retarded potentials and radiation.

4. The beauty of deductive thinking
   The conceptual development of much of the course content was repeatedly shown to hinge on just two basic principles: the covariance of equations and the existence of a finite speed of propagation.

5. Introducing tensors
   Tensors are typically regarded as “very abstract” and alienating to beginners, partly because they are often introduced without much context. However, one can motivate them naturally: physical theories are largely concerned with differential equations, and the demand that the same laws hold across inertial frames — i.e., covariance of equations — naturally leads to the introduction of covariant and contravariant tensors, quantities that transform in such a way as to preserve the form of differential equations under coordinate transformations. Of course, it was stressed that one has to solve enough problems to become truly familiar with them.

6. Introducing the notion of fields
   This is probably the first course in the curriculum where students truly encounter the notion of “fields” in its full extent. They calculate and see that these are extended objects spread out in space and time, possessing their own dynamics. Moreover, they carry energy and momentum, much like the classical particles familiar from the Newtonian context.

7. The relativistic nature of Maxwell’s equations
   Maxwell’s equations, though historically written before the formulation of relativity, are fundamentally compatible with special relativity and acquire a manifestly Lorentz-covariant form when written in spacetime language. In fact, Einstein’s early concern was precisely that Maxwellian electrodynamics was incompatible with Galilean kinematics, which led him to search for a broader framework. Relativity is not something exotic, nor is it relevant only to high-energy physicists. The familiar Maxwell equations, electromagnetic waves, radiation, and related phenomena are already relativistic in character.

8. For advanced students: field theory, gauge invariance, and unification
   For advanced students, this is the first time they encounter the construction of a relativistically covariant classical field theory. Landau presents that construction wonderfully. It is also the first time they encounter the construction of a gauge theory and come across the concept of gauge invariance. Further, it is the first time they see symmetry-based unification of seemingly disparate concepts such as space and time, or electric and magnetic fields. It was stressed, whenever appropriate, that these principles would remain important far beyond this course, especially for theoretically inclined students.

Most of the time, the relevance of core or foundational courses such as Classical Mechanics and Electrodynamics becomes obscure because of how they are taught, often leading students to ask, “Why study topics like these?” Students genuinely appreciate it when they are shown why certain courses are considered foundational or fundamental, and when the broad themes running through them are properly contextualized.